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Vertical velocities associated with gravity waves measured in themesosphere and lower thermosphere with the EISCAT VHF radar

N. J. Mitchell, V. St. C. Howells

Department of Physics, The University of Wales, Aberystwyth, Ceredigion, SY23 3BZ, UK (E-mail:[email protected])

Received: 13 October 1997 /Revised 30 April 1998 /Accepted: 15 May 1998

Abstract. The EISCAT VHF radar (69.4°N, 19.1°E)has been used to record vertical winds at mesopauseheights on a total of 31 days between June 1990 andJanuary 1993. The data reveal a motion ®eld dominatedby quasi-monochromatic gravity waves with represen-tative apparent periods of �30±40 min, amplitudes of upto �2.5 m s)1 and large vertical wavelength. In someinstances waves appear to be ducted. Vertical pro®les ofthe vertical-velocity variance display a variety of forms,with little indication of systematic wave growth withheight. Daily mean variance pro®les evaluated forconsecutive days of recording show that the generalshape of the variance pro®les persists over several days.The mean variance evaluated over a 10 km height rangehas values from 1.2 m2s)2 to 6.5 m2s)2 and suggests asemi-annual seasonal cycle with equinoctial minima andsolsticial maxima. The mean vertical wavenumber spec-trum evaluated at heights up to 86 km has a slope(spectral index) of )1.36 � 0.2, consistent with obser-vations at lower heights but disagreeing with thepredictions of a number of saturation theories advancedto explain gravity-wave spectra. The spectral slopesevaluated for individual days have a range of values, andsteeper slopes are observed in summer than in winter.The spectra also appear to be generally steeper on dayswith lower mean vertical-velocity variance.

Key words. Meteorology and atmospheric dynamics(middle atmosphere dynamics; waves and tides)

1 Introduction

Gravity waves play an important role in the dynamics ofthe middle atmosphere. The waves are now known to

transport energy and momentum from their lower-atmosphere sources and deposit them at greater heights.Modelling studies have revealed that the verticallytransferred horizontal momentum profoundly in¯uencesthe circulation of the mesosphere/lower-thermosphere(MLT) region, decelerating the atmospheric zonal ¯owaway from radiative-equilibrium values, closing themesospheric jets and inducing a meridional circulationwhich ultimately cools the summer mesopause toextremely low temperatures through the consequentascent of air required by continuity considerations (e.g.Houghton, 1978; Lindzen, 1982; Holton, 1982, 1983;Matsuno, 1983; Dunkerton, 1982; Miyahara, 1984;Holton and Zhu, 1984; Garcia, 1989; McIntyre, 1989).The turbulence arising from gravity-wave breakingprocesses is also important in the transport of heat andconstituents. The temporal and geographic variations ingravity-wave ¯uxes thus contribute to the variability oflarge-scale properties of the atmosphere. Attempts tounderstand and model the middle atmosphere thereforerequire detailed information about properties of thegravity-wave ®eld.

The majority of recent studies of gravity waves in themiddle atmosphere have been made primarily at mid-latitudes and have employed either Rayleigh andsodium lidars to measure perturbations of temperatureand density (e.g. Gardner et al., 1989; Mitchell et al.,1991; Wilson et al., 1991; Meriwether 1993; Senft et al.,1993; Meriwether et al., 1994) or MF, MST andincoherent scatter radars to measure wind perturbations(e.g. Vincent and Reid, 1983; Meek et al., 1985a, b;Vincent and Fritts, 1987; Fritts and Yuan, 1989; RuÈ sterand Reid, 1990; Fritts et al., 1992; Nakamura et al.,1993; Gavrilov et al., 1995). However, despite theconsiderable body of these latter radar observations,the great majority of such studies have concernedmeasurements of horizontal winds only. The fact thatthe vertical-wind perturbations associated with gravitywaves are usually an order of magnitude smaller thanthe horizontal winds makes these measurements techni-cally di�cult, and even at MLT heights the verticalCorrespondence to: N. J. Mitchell

Ann. Geophysicae 16, 1367±1379 (1998) Ó EGS ± Springer-Verlag 1998

motions are expected to have amplitudes of only a fewm s)1 at most. An additional limitation is the verylimited temporal and spatial coverage o�ered by someof the techniques able to make such measurements.Consequently, gravity-wave vertical motions at MLTheights have been investigated by only a very limitednumber of studies (e.g. Reid, 1987; Meek and Manson1989; Fritts et al., 1990; Fritts and Hoppe, 1995; Hoppeand Fritts, 1995).

Gravity waves occur at scale sizes too small to beresolved in current GCMs and so the parameterisationsof the gravity-wave ®eld is an important problem forsuch models encompassing the MLT region (e.g. Hines,1997a, b). A key aspect in such parameterisations is anunderstanding of the nature of gravity-wave spectra andof the physical processes shaping the spectra. Gravity-wave spectra have been known for a number of years tohave a simple ``universal'' power-law form at verticalwavenumbers above a critical value (VanZandt, 1982;Smith et al., 1987). The spectral region over which thepower-law form holds is taken to represent saturatedwaves and numerous observations have revealed a formin which the power spectral density of horizontal windsexpressed in terms of vertical wavenumber, Fu(m), varieswith vertical wavenumber, m, as Fu(m) µ m)3 (VanZ-andt, 1982).

A number of theories invoking di�erent physicalmechanisms have been advanced to account for theobserved spectra, including theories based on linearinstability (Dewan and Good, 1986; Smith et al., 1987),di�usive damping (Weinstock, 1990), Doppler spreading(Hines, 1991), saturated-cascade processes (Dewan,1994) and di�usive ®ltering (Gardner, 1994). Thesetheories are all successful in accounting for the observedspectral amplitudes and slopes when the spectra arethose of ¯uctuations in horizontal velocity or temper-ature, and so observations of these quantities cannot beused to distinguish between the theories. However, thevertical-wavenumber spectra of vertical velocities pre-dicted by at least some of the theories for saturatedwaves are signi®cantly di�erent, and so such observa-tions can in principle be used as a test of saturationtheories (for example, see the extensive discussion ofGardner, 1996).

In this study we report observations of the vertical-velocity ®eld made using the EISCAT VHF incoher-ent-scatter radar. Previous studies of the verticalmotions associated with gravity waves made using thisinstrument have employed measurements of very hightime resolution, but of durations limited to a fewhours (e.g. Fritts et al., 1990; Fritts and Hoppe, 1995;Hoppe and Fritts, 1995). In the present study a muchlarger data set of lower time-resolution measurementsmade on a total of 31 days is used to investigate anumber of properties of the vertical motions due togravity waves at mesopause heights. The generalcharacter of the waves observed are described andthe variance of vertical velocities is used as a tracer ofwave activity across the height range observed. Spectraof vertical velocities in terms of vertical wavenum-ber are presented and the availability of data from

di�erent seasons, and from short runs of consecutivedays of recording, allows an investigation of theseasonal and day to day behaviour of some aspectsthe vertical-velocity ®eld.

2 Measurements and data analysis

The EISCAT VHF radar used in this study is located at69.4°N, 19.1°E, near Tromsù in Norway. The radaroperates at a frequency of 224 MHz and has a meanpower output of about 0.3 MW. The data were recordedwith EISCAT Common Program 6 which uses a singlevertical beam and can measure vertical winds betweenabout 74±97 km, however, no information is providedabout horizontal winds or temperatures. The rangegates used correspond to a vertical resolution of 1.05km and the data were pre-integrated in time to givesamples every 5 min. The lowest height gate corre-sponded to a height of 75.0 km, and usable signal wasrarely returned above about 97 km, so that at mostabout 21 consecutive height gates of data were avail-able. In each case, the quality of the ®t to the returnedspectrum was used to provide a value for the uncer-tainty associated with each velocity determination. Inthe work reported here the decision was made to rejectall data points with velocity uncertainties in excess of0.5 m s)1.

Vertical winds were recorded on a total of 31 daysbetween June 1990 and January 1993. Most of data formshort series of several consecutive days of observation.The lengths of the continuous sections of data recordingwere 6, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1,1 days. The dates of theindividual observation sessions are given in Table 1. Notall the days of observation recorded signal satisfying theabove criterion in all height gates at all times, andTable 1 also indicates for each day the percentage of the6912 possible data points over 24 h between 75 and 100km which yielded usable data. From the values inTable 1, it is clear that some days of observation yieldeda comparatively limited number of data points. Ingeneral, this was due to curtailment of recording outsideof a limited number of hours in the particular day, ratherthan patchy data throughout the day.

3 General characteristics of the vertical motion ®eld

Examples of the vertical-velocity ®elds recorded arepresented in Fig. 1a, b as time-height contour plots. No®ltering in time or height has been applied to the data,which were recorded over the two continuous 24-hperiods covering 31 July 1992 (Fig. 1a) and 18 March1991 (Fig. 1b). Note that the ®gures are plotted ondi�erent velocity scales in order to highlight details ofeach data set. Both sets of data display generalcharacteristics typical of all the wave ®elds recorded inthis study.

Considering the data for 31 July 1992, the motion®eld is seen to be dominated by short-period oscilla-tions, with only limited indications of longer-period

1368 N. J. Mitchell, V. St. C. Howells: Vertical velocities associated with gravity waves

motions. An example of the latter is the suggestion of adescending phase front at heights between about 88±96km from about 06:00 to 12:00 UT. The dominant,shorter-period motions appear to be quasi-monochro-matic gravity waves with amplitudes of up to �2.5 m s)1

and periods of 30±40 min. The wave events appear tohave durations of up to about 8 h (e.g. the quasi-monochromatic wave maximising at heights near 84km). In all cases the vertical wavelengths appear to bevery large, with the phase fronts being, in most cases,e�ectively vertical. The vertical coherence of the mo-tions is striking, with individual phase fronts appearingover vertical extents of some 10 km or so. Signi®cantly,wave amplitudes do not appear to increase systemati-cally with height, and, except in isolated episodes, thereis little sign of the e(z/2H), amplitude growth associatedwith energy-conserving motions. Wave activity appearsto be highly variable over both the height and time rangeobserved. For instance, compare the strong waveactivity centred at heights near 84 km at 09:00 UT,where velocities are recorded of up to �2.5 m s)1, withthe quiescent wave ®eld between �88±94 km at the sametime, where amplitudes are <1 m s)1.

The data for 18 March, 1991 show a broadly similarpattern of vertical phase fronts. However, the ampli-tudes are signi®cantly reduced compared to those for 31July, 1992 except for larger amplitude motions at allheights before about 02:00 UT, and the descendingregion of higher amplitudes starting above 95 km at thestart of the day and slowly descending after about 10:00UT to �90 km by 20:00 UT.

Table 1. Dates of EISCAT VHF Common Program 6 verticalwind data. The percentage of usable data points is indicated aftereach date

1990June 12 (53%), 13 (22%)July 30 (21%), 31 (74%)August 1 (10%), 27 (34%), 28 (30%)September 20 (18%)November 20 (15%), 21 (38%)

1991February 12 (10%), 13 (54%), 14 (7%), 20 (19%), 21 (25%)March 17 (30%), 18 (84%), 19 (70%), 20 (37%)June 4 (52%)

1992July 30 (37%), 31 (83%)October 27 (34%), 28 (73%), 29 (51%)

1993January 20 (10%), 21 (5%), 22 (53%), 23 (42%), 24 (52%),

25 (22%)

Fig. 1. Time-height contours of vertical velocity for data recorded on a 31 July, 1992, and b 18 March, 1991. Note the di�erent velocity scalesused

N. J. Mitchell, V. St. C. Howells: Vertical velocities associated with gravity waves 1369

This pattern of motion is in marked contrast to thatof horizontal winds revealed throughout the meso-sphere by radar wind measurements (e.g. Manson,1990; Nakamura et al., 1993), throughout the upperstratosphere by lidar measurements of temperature anddensity perturbation (e.g. Gardner et al., 1989; Mitchellet al., 1991; Wilson et al., 1991) and throughout thelower stratosphere by lidar and MST/ST-radar mea-surements (e.g. Meriwether, 1993; Mitchell et al., 1994).In these measurements the motion ®eld is found to bedominated by quasi-monochromatic waves with periodscomparable to the inertial period and vertical wave-lengths ranging from �2 km in the lower stratosphere to�15±20 km at mesopause heights.

In a number of the longer continuous data intervalsconsidered here there are episodes in which a waveappears con®ned to a particular height range forseveral hours and has an amplitude which becomesvery small or zero at the edges of this range. Anexample can be seen in Fig. 1a from about 06:00 to13:00 UT centred at heights of �83 km. The data forthis event is presented in more detail as a time-heightcontour map of vertical velocity for times between06:00±13:00 UT and heights between �77±90 km inFig. 2. Of particular interest is the way in which thewave's amplitude appears to decrease rapidly from amaximum of about 2.5 m s)1 at �83 km to very smallamplitudes near �88 km.

To investigate this behaviour further, amplitudes ineach height gate were calculated by least squares®tting a sine wave to the six-hour time intervalspeci®ed. For each height gate the period of the ®ttedfunction was varied until a maximum in amplituderesulted. The wave amplitudes for each height gatederived by this method are presented as a function ofheight in Fig. 3. Figure 3 clearly shows the rapid riseand curtailment of wave amplitude with height, withwave activity essentially con®ned between heights of�78±87 km.

Also indicated on Fig. 3 are four predicted wave-amplitude growth curves, initiated from the measuredwave amplitude of 0.6 m s)1 at a height of 77.7 km. The

growth curves are calculated assuming a simple e(z/2H)

amplitude growth in an isothermal atmosphere at theindicated temperatures of 180, 160, 140 and 120 K. Itcan be seen from Fig. 3 that the wave grows with heightmore rapidly than would be the case for a conservativemotion in an atmosphere at any credible mesopause-region temperature. This, combined with the verticalphase coherence strongly indicates that the wave is infact ducted. Unfortunately, no information is availableabout the temperature or horizontal wind structure ofthe atmosphere for the data considered here, so thesuggestion of ducting cannot be directly con®rmed forthis example.

It has recently become recognised that the middleatmosphere provides an environment rich in the tem-perature and wind structures capable of ducting gravitywaves. Ducts can be formed by either Doppler orthermal processes. In the case of Doppler ducts,Doppler shifting increases a wave's intrinsic frequencyto a value above that of the Brunt-Vaisala frequency,N, outside of the duct. Chimonas and Hines (1986)described the behaviour of Doppler-ducted gravitywave and reported several examples of waves appar-ently so ducted at tropospheric heights. Similarly,thermal ducts may be formed when ¯uctuations of Nwith height, perhaps caused by wave-induced pertur-bations of the background temperature gradient, trap awave within a layer of atmosphere where at immedi-ately greater and lesser heights the value of N is lowerthan the wave frequency. Lidar observations have infact revealed temperature lapse rates approaching theadiabatic in the upper mesosphere (e.g. Wilson et al.,1991; Meriwether et al., 1994), and as temperature lapserates approach the adiabatic, the associated value of Ntends towards zero, so gravity waves of even middleperiods can become trapped in a duct bounded by suchregions.

Radar studies of the vertical winds associated withgravity waves have usually used MST or ST radars andfocused on tropospheric or lower stratospheric heights.A comparatively limited number of such studies havebeen made of waves at mesopause heights. In this latter

Fig. 2. Time-height contours ofvertical velocity for data recordedon 31 July, 1992 between 06:00±13:00 UT at heights of 76±90 km


group are studies of summer mesopause-region verticalwinds made using the EISCAT VHF radar by Frittset al. (1990), in which vertical winds measured with a 10s time resolution were recorded for a total of 36 h onnine days in June and July 1987. Data were also used intheir study from 3 days of recording with the SOUSYVHF radar. Hoppe and Fritts (1995) and Fritts andHoppe (1995) again used EISCAT VHF data to measurevertical mesopause-region winds on one day in June1988, this time under PMSE conditions. The strongscatter from the PMSE allowed a time resolution of 2 s.The data were recorded in two intervals, one of about3 h, the other of 1 h following after a gap of 1 h. In thesestudies, wave motions with similar periods, verticalcoherence and vertical wavelengths to those reportedhere were identi®ed by the authors in at least some of thedata.

Our observations, based on a longer data setcovering winter and equinoctial conditions as well assummer, suggest that this pattern is representative of themesopause region throughout the year. However, thereare some di�erences apparent in comparing the high-resolution studies described above with our observa-tions. In particular, the study of Fritts et al. (1990)found vertical velocities up to �10 m s)1, a factor 3greater than recorded in our lower resolution data. Thehigher velocities may simply re¯ect a particularly activeday, or may be a product of the di�erent timeresolutions. Also, an example of high-frequency waves(wave periods �1±10 min) presented by Hoppe andFritts (1995), where the authors interpret the verticalphase coherence over some 7.2 km of altitude aspossible evidence of ducting, does not display theapproximately Gaussian pro®le of amplitude withheight evident in our data for the event described aboveand displayed in Fig. 2 and 3.

Further evidence that ducted motions may becommon at mesopause heights has been provided bythe combined MF radar and all-sky nightime airglowCCD imager study reported by Taylor et al. (1995), inwhich the authors demonstrated that small-scalewaves exhibiting highly coherent phase structuresobserved at four di�erent emission altitudes wereDoppler ducted.

4 Vertical pro®les of velocity variance

Variance of the vertical velocity in each height gateprovides a measure of wave activity. The vertical pro®leof variance can be used as a tracer to give insight intophysical processes within the wave®eld. For each of the31 days of data, the mean pro®le of vertical-velocityvariance was calculated. A variety of di�erent types ofvariance pro®le are observed, and Fig. 4a±d presentsfour representative examples of vertical pro®les ofvariance between 76±95 km from 4 June 1991, 31 July1990, 27 August 1990 and 25 January 1993, respectively.In each case, all the data recorded in the indicated 24-hinterval are considered, so the curves incorporatecontributions from all waves present. It is clear fromFig. 4 that a wide variety of forms of variance curve areobserved.

The example for 4 June 1991 (Fig. 4a) reveals aquiescent wave ®eld with approximately constant vari-ance values �1 m2s)2 at all but the greatest heights. Theexample from 31 July 1990 (Fig. 4b) also reveals a wave®eld which demonstrates little growth with height,except above �91 km, however, in this case the wave®eld is considerably more energetic with variances of�3±5 m2s)2 over most of the height range. A signi®-cantly more energetic wave ®eld is revealed in the datafor 27 August 1990 (Fig. 4c), where the varianceincreases in an approximately linear fashion from valuesof �2 m2s)2 near 76 km to �15 m2s)2 near 95 km. Amore complicated structure is displayed in the data for25 January, 1993 (Fig. 4d) in which the variance risesfrom values near 3 m2s)2 at 76 km to a maximum of �12m2s)2 near 89 km then decreases to values near 2 m2s)2

near 95 km.In two of the short series of consecutive days of

recording it was noticed that the vertical pro®les ofvertical wind variance tended to have a similar shapeover a time scale of days. An example is presented in theseries of four daily mean variance curves calculated foreach day in the period 17±20 March 1991 shown inFig. 5a±d. In these four pro®les a region with variancevalues ¯uctuating between �2±5 m2s)2 extends toheights of about 86 km with little indication of anysystematic growth, above which the variance increasesrapidly with height. A similar sequence of pro®les forthe period 21±25 January 1993 is presented in Fig. 6a±e,in which variances rise from values of about 5 m2s)2 orless to values of >10 m2s)2 at heights near 90 kmbefore declining again at greater heights. In each ofthese two series the general shape of the variancepro®les remains similar from day to day. A possible

Fig. 3. The amplitude-height pro®le for the quasi-monochromaticgravity wave observed between 06:00±12:00 UT on 31/7/92. Alsoindicated are the predicted amplitude growth curves for temperaturesof 180, 160, 140 and 120 K isothermal atmospheres


reason for this behaviour is the background structure ofwind and/or temperature, but in the absence of thisinformation it is not possible to investigate the phe-nomenon fully.

The mean variance over an arbitrary height range,evaluated as a function of time, can be used toinvestigate the seasonal behaviour of wave activity.The averaging over a height range is necessary becauseof the small-scale ¯uctuations in the vertical pro®les ofvariance when this quantity is evaluated as a dailymean for each height gate. Figure 7 presents the mean

variance over the height interval 76±86 km evaluatedfor each of the 31 days of observation plotted againstday number in the year. The upper limit was selectedbecause some of the vertical pro®les of variancechange character at about this height, and becausethis is representative of mesopause heights abovewhich it might be anticipated that the environmentin which the waves are propagating changes. Datafrom all four years of observation are superimposed.The limited number of samples, and the small fractionof usable data from some days of observation,

Fig. 4. Pro®les of the daily-mean variance of vertical velocity for a 4 June, 1991; b 31 July, 1990; c 27 August, 1990 and d 25 January, 1993


introduces uncertainty into any attempt to discern aseasonal cycle. However, there is a suggestion of awinter maximum with mean variances having valuesup to 6.5 m2s)2. Values as low as 1.2 m2s)2 areevident around the spring equinox, and a secondarymaximum with values up to 5.8 m2s)2 is evident nearthe summer solstice. The situation at the autumnalequinox is less clear, with three data points fromOctober 1992 (day numbers 299, 300, 301) suggestingthat a secondary minimum may occur near theautumnal equinox.

A limited number of other studies have measuredthe vertical pro®le of vertical-velocity variance atmesopause heights. Fritts and Yuan (1989) used thePoker Flat MST radar over 8 days in July 1986 andfound vertical-velocity variances to be approximatelyconstant over the height range 82±87 km, with valuesof �6 m2s)2 below 87 km. RuÈ ster and Reid (1990) usedtwo days of data recorded by the SOUSY VHF radarin June 1987 at heights between �83±90 km, and foundvertical motions dominated by oscillations with periodsof less than one hour and with variances which

Fig. 5 a±d. Pro®les of the daily-mean variance of vertical velocity for four successive days in 1991. a 17 March, b 18 March, c 19 March and d 20March


gradually increased from �1 m2s)2 near 83 km to �3m2s)2 near 90 km. The study of Fritts et al. (1990)reported variances increasing from �3.5 m2s)2 at 81.6km to �6 m2s)2 at 90 km with a mean of 5.2 m2s)2 forthe 36 h of measurements made with the EISCAT-VHF radar (selecting data points with errors of £0.3m s)1), and variances ranging from �4 m2s)2 at 83 kmto �6 m2s)2 at 90 km with a mean of 5.1 m2s)2 for thethree days of measurements made by the SOUSYradar.

The variances presented in this study are thusbroadly comparable with this earlier work, although it

should be noted that the highly variable wave ®eldsdescribed in Sect. 3 and illustrated in Fig. 1a, b suggestthat measurements of short duration made only a fewhours apart could produce quite di�erent pro®les andmeans of vertical-velocity variance, so direct compari-sons should be treated with caution. The persistentshape of the daily mean variance pro®les over a numberof days is similar to the report of persistent horizontalmomentum ¯ux pro®les of Fritts et al. (1992) whoexamined 16 days of data recorded from heights of �68±84 km using the Jicamara MST radar (12°N) duringsummer 1987. A correspondence in momentum ¯ux and

Fig. 6 a±e. Pro®les of the daily-mean variance of vertical velocity for four successive days in 1993. a 21 January, b 22 January, c 23 January and d24 January e 25 January


variance pro®les is to be expected as it is believed to bethe high-frequency waves which make the dominantcontribution to the momentum ¯ux, and it is also thesewaves which make the greatest contribution to thevertical-velocity variance.

A number of sophisticated climatologies of gravitywaves at mesopause heights have been constructedfrom middle or low-latitude measurements of horizon-

tal winds. Climatologies of gravity wave motions athigh latitude are much less well developed. MF-radarstudies based on horizontal winds have indicated thatat mid-latitudes there is an annual cycle for long-period gravity waves with a winter maximum, whilst athigher wave frequencies (e.g. periods 0.2±2.5 h) amarked semi-annual component with a secondarymaximum in summer becomes apparent (e.g. Mansonand Meek, 1993; Gavrilov et al., 1995; Nakamura et al.,1996).

Our results for the seasonal variation in variance arenot in disagreement with the idea that the vertical-velocity ®eld may behave in a similar manner to that ofhigh-frequency gravity waves observed in horizontalvelocity measurements. Such a similarity is to beanticipated given the dominant contribution to thevertical velocities made by high-frequency waves (Sect.3) but requires a larger data set than currently availablefor certainty.

5 Vertical-wave number spectra

The vertical-velocity pro®les were used to investigatethe spectral composition of the wave ®eld in terms ofpower spectral density as a function of verticalwavenumber, Fw(m). In calculating the spectra onlydata up to heights of 86 km were considered. Thisarbitrary upper height limit was chosen in an attemptto avoid spanning the mesopause for the reasons givenin Sect. 4, and this restriction limited the number ofheight gates available to at most 10 (i.e. a 10.5 kmheight range). In a signi®cant fraction of the data,fewer points were available for analysis because ofepisodes of low signal-to-noise in the records. Ratherthan interpolate over gaps in the individual pro®les ofvertical velocity, a Lomb-Scargle periodogram wasused. This technique allows the spectral analysis ofnon-periodically spaced data. Spectra were calculatedusing the periodogram for each individual verticalpro®le of vertical velocity, then averaged to give adaily-mean spectrum. No attempt was made to try anddivide data into cases with and without ducted waves.In the absence of horizontal wind and/or temperatureinformation only the most conspicuous cases (such asthat of Figs. 2 and 3) could be identi®ed. The meanspectrum calculated from all the available data in which9 or 10 height gates satis®ed the velocity uncertaintycriterion is shown in Fig. 8. Notice that in Fig. 8 thereare data points corresponding to spectra calculatedover a vertical extend of both 10.5 km (10 height gates)and 9.45 km (9 height gates).

If the mean spectrum is assumed to have a power-lawform, Fw(m) µ mk, the spectral index or slope, k, is foundby a linear regression to be )1.36 � 0.2. However, theindividual daily mean spectra display a considerablerange of spectral slopes from �)0.41 to �)2.25.

An indication of the range of spectral indices is givenin Fig. 9, which presents the daily-mean spectral indexagainst day number; again, data from di�erent years are

Fig. 6e.

Fig. 7. The daily-mean vertical velocity variance over the heightrange 76±86 km for all data


superimposed. Figure 9 shows a clear trend for thesteeper spectral indices to be found in summer andshallower slopes to occur in winter. The equinoctial

values are comparable with those of the winter months,although the small number of days of recording preventsde®nite conclusions. Finally, it should be noted that thewavenumber regime accessible to our analysis is likely tofall entirely within that composed of saturating waves.Typical values for m* (the transitional wave numberbetween saturated and unsaturated waves) reported instudies of horizontal winds at mesopause heightscorrespond to vertical wavelengths in excess of 15 km(e.g. Smith et al., 1987).

Finally, we examined the spectral index as a functionof the daily-mean variance over the 76±86 km heightrange described in Sect. 4. The variation of spectralslope with daily-mean variance is shown in Fig. 10.From this plot it can be seen that there is a tendency forthe spectral index to become more positive (i.e. shal-lower spectral slopes) on days when the mean variancehas higher values.

Observations made at lower heights have also investi-gated spectral slopes at high wavenumbers. The ST-radarstudies of Kuo et al. (1985) measured the slopes ofvertical-velocity spectra to be near )1.7 in the middletroposphere and near )0.34 in the lower stratosphere.Larsen et al. (1987) reported slopes near )1.3 fortroposphere/lower-stratosphere data and Sidi et al.(1988) reported a spectral slope of )1.8 for the lowerstratosphere based on a high-resolution balloon mea-surement. At mesopause heights, Fritts and Hoppe(1995) measured slopes of from �)1 to �)3 for twoshort duration, high-resolution measurements using theEISCAT VHF radar. Gardner et al., (1995) used aDoppler/Sodium lidar to measure vertical wind pro®lesover 6 hours between heights of 84±100 km andmeasuredthe spectral slope to be )1.4; however, in this study the

Fig. 8. The mean vertical-wavenumber spectrum of all data over therange 76±86 km. Note that data from spectra calculated with either 9or 10 data points are included

Fig. 9. The daily-mean spectral index (slope) for each of the days ofdata evaluated over the height range 76±86 km

Fig. 10. The spectral index (slope) as a function of daily-meanvariance over the height range 76±86 km


data was integrated for 20 minutes before spectralanalysis, which may have supressed the amplitudes ofany high-frequency motions present if of similar periodsto those reported here.

A major motivation in studying the vertical-velocityspectra of atmospheric gravity waves is that they o�er anopportunity to test theories which have been advanced toexplain the observed ``universal spectrum'' of gravity-wave horizontal motions ®rst reported by VanZandt(1982). Gardner (1996) reviewed current theories in somedepth, and compared their predictions for a number ofatmospheric parameters, including the form of thevertical-wave number spectrum of vertical velocities,Fw(m). This parameter is particularly important because,unlike the spectra of horizontal velocities, at least someof the theories predict di�erent behaviour and so allow atest against observation.

The theories invoking linear convective or dynamicinstability (Dewan and Good, 1986; Smith et al., 1987)predict a fully separable (m, x) spectrum in which theshapes of the horizontal and vertical-velocity wavenumber spectra, Fu(m) and Fw(m), respectively, will beidentical within the saturated part of the wave spectrum(that which lies at m-values above the critical value m*).There is a considerable, but not unanimous, body ofevidence that suggests the Fu(m) spectra have slopesclose to )3 (e.g. Smith et al., 1987), and so similar valuesare predicted by this theory for the Fw(m) spectra. Thedi�usive ®ltering theory of Gardner (1994) predicts thatthe (m, x) spectrum of horizontal winds is not separableand that the vertical-velocity spectra, Fw(m), will have apower-law form with slopes µ m(5-2p), where p is thespectral index of the frequency spectrum of horizontalwinds, and has observationally based values of p �1.6±2. Thus the di�usive ®ltering theory predicts Fw(m) µ mto Fw(m) µ m1.8, i.e., positive spectral slopes of steepness+1 to +1.8. The saturated cascade theory of Dewan(1994) predicts non-separable (m, x) spectra with slopesof about +1. Neither the di�usive damping theory ofWeinstock (1990), nor the Doppler spreading theory ofHines (1991) are as yet su�ciently developed as to beable to make predictions of the vertical-velocity spectralform.

The results presented in our study suggest that theobserved spectral slopes of the vertical-velocity spectraare not close to those predicted by any of the theoriesdescribed already. At all times the spectra have anegative slope, but usually have values between �)0.4to �)1.6; signi®cantly less than predicted by the linearinstability theories. However, a complication in thestudy of the vertical motion ®eld is the probablepresence of a population of ducted waves. In caseswhere there is a signi®cant contribution to the verticalvelocities from ducted waves, the spectra may becontaminated by the approximately Gaussian pro®leof amplitude across the duct (Fig. 3). An assessment ofthe prevalence of ducted motions is thus essential beforethe wavenumber spectra of vertical velocity can be usedas a de®nitive test of saturation theories.

6 Conclusions

We have presented analysis of 31 days of vertical-windrecords made using the EISCAT VHF radar between 12June, 1990 and 25 January, 1993. The motion ®eldappears to be dominated by high-frequency motions oflarge vertical wavelength and periods of order 30±40min. In at least some of the data there are strongindications that waves are trapped in ducts, althoughadditional information about horizontal winds and/ortemperatures would be necessary to con®rm thissuggestion. Such additional information would also beuseful in investigating the possible e�ects of Dopplershifting on the observed waves. The vertical-velocityvariances display various forms, but there is littleindication of the expected wave growth with height,despite the fact that the dominant waves appear to havevery large vertical wavelengths, and so might beexpected to belong to the unsaturated part of the spec-trum. The persistence of the general form of the verticalpro®les of daily mean variance suggests that the wave®eld is being shaped by longer-lasting properties of theatmosphere, either the background temperature ormean (horizontal) winds, and is similar to that reportedin at least one study of the vertical ¯ux of horizontalmomentum (Fritts et al., 1992). The seasonal variationin vertical-velocity variance appears to be consistentwith the semi-annual cycle reported in studies of short-period gravity waves made using observations ofhorizontal velocity. The spectral properties of the wave®eld examined over a wave number range which wouldbe expected to be composed of saturated waves agreeswith observations made in the troposphere and lower-stratosphere, but do not agree with the predictions oflinear instability theory (Dewan and Good, 1986; Smithet al., 1987), di�usive-®ltering theory (Gardner, 1994) orsaturated-cascade theory (Dewan, 1994). However, thepresence of ducted motions within the wave ®eld mayseriously contaminate the vertical-wavenumber spectra.This may be an explanation of why the spectra haveshallower slopes for wave ®elds of higher variances(Fig. 10) and again highlights the need for studies inwhich simultaneous measurements are made of severalatmospheric parameters.

Acknowledgements. We are indebted to the Director and sta� ofEISCAT for operating the facility and supplying the data. EISCATis an International Association supported by Finland (SA), France(CNRS), the Federal Republic of Germany (MPG), Japan (NIPR),Norway (NFR), Sweden (NFR) and the United Kingdom(PPARC).

Topical Editor D. AlcaydeÂ thanks N. F. Arnold for his help inevaluating this paper.

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